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Examples of Jacobi forms in many variables

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Jacobi modular forms: 30 ans après

National Research University Higher School of Economics

4.8 (12 classificações) | 2.5K alunos inscritos

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This is a master course given in Moscow at the Laboratory of Algebraic Geometry of the National Research University Higher School of Economics by Valery Gritsenko, a professor of University Lille 1, France. Jacobi forms are holomorphic functions in two complex variables. They are modular in one variable and abelian (or double periodic) in another variable. The theory of Jacobi modular forms became an independent research subject after the famous book of Martin Eichler and Don Zagier “Jacobi modular forms” (Progress in Mathematics, vol. 55, 1985) which was cited more than a thousand times in research papers. This is due to many applications of Jacobi forms in arithmetic, topology, algebraic and differential geometry, mathematical and theoretical physics, in the theory of Lie algebras, etc. The list of mentioned subjects shows that my course might be useful for master and Ph.D. students working in different directions. Motivated undergraduate students can also study this subject. To follow the course one has to know only elementary basic facts from the theory of modular forms (for example, the paragraphs 1-4 of the chapter VII of Serre’s “A Course in Arithmetic” are enough). The main hero of the course is the Jacobi theta-series. Using it we will construct a lot of concrete examples of Jacobi forms in one or many abelian variables, in particular, Jacobi forms for root systems. For some of you, who will be successful with the theoretical exercises of the course, I am ready to formulate research problems for Master or Ph.D. thesis. (Ph.D. support might be available at CEMPI in Lille or at the Faculty of Mathematics of National Research University Higher School of Economics in Moscow)

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Na lição

Jacobi forms in many variables and the Eichler-Zagier Jacobi forms

This module is devoted to Jacobi forms in many variables. In this module we also define classical Eichler-Zagier Jacobi forms in terms of Jacobi forms in many variables. Also there is a peer review in the end of this module.

Jacobi theta-series and Jacobi forms in many variables15:03

Jacobi theta-series and Jacobi forms in many variables (part 2)15:57

Hyperbolic reformulation14:18

Examples of Jacobi forms in many variables14:55

Examples of Jacobi forms in many variables (part 2)16:20

Examples of Jacobi forms in many variables (part 3)16:29

Ministrado por

Valery Gritsenko

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My next remark

about the place

of Eichler-Zagier Jacobi form in this definition.

We start our course,

With Jacobi form in one variable.

Now we extended our main definition but this is really an extension,

a generalization or a new definition.

An Eichler-Zagier Jacobi form the serial of this

function was constructed in the famous book of

Eichler-Zagier published exactly 30 years ago.

This is Jacobi form of weight K and index m.

I put here Eichler-Zagier to emphasize that we have a Jacobi form

in one abelian variable.

We have no lattice in this definition but

if we analyze the functional equation,

We see then the functional equation

of Jacobi form of Eichler-Zagier

we had here the factor e to

the power pi i pi m z to the square.

What does it mean?

Now I write this

factor once more.

So the factor in the functional equation (M),

e to the power 2 pi i m the index c

z to the square over c tau + d.

In our functional equation with the lattice,

this is e to the power pi i c z to the square, c tau d.

It means that the lattice

in the definition

of Eichler-Zagier,

Is the lattice 2 or A1.

This is the lattice generated by

a vector u with square 2.

Means in the Jacobi

form of Eichler-Zagier of weight k and index m

are Jacobi form, according to our new definition, of weight k

for the lattice A1 over the lattice 2 and index m.

Or if you like we can take

the lattice 2m and index 1.

So, index M corresponds to the lattice,

2m or the lattice A1.

This is the standard,

the simplest root lattice renormalized by m.

So this is a place of the classical Eichler-Zagier Jacobi form

in our definition.

Moreover, we discussed the graded ring and

in Eichler-Zagier,

They determined the structure of

the graded ring of the weak

Jacobi form for the lattice A1.

They proved that this graded ring over

the ring of the graded ring of modular form

with respect to SL2 z has only three generators.

This is Jacobi form of weight -2, 1,

the principal generator we constructed in the beginning of this lecture.

Then Jacobi form of weight 0 and index 1 and

the search function -2, 1.

These two functions we saw in

the beginning of our lecture.

Let me show them for you once more.

This is the first function.

And the second function, you can get from this function,

phi- 1, 2.

This is theta 2 z

over eta q and tau.

So you see that the theta function gives

us the main generators of the graded

ring of the weak Jacobi cusp form.

In the case of Eichler-Zagier, the case in one variable without any lattice.

This is a case of the simplest possible root system, A1.

Now, I would like to construct new examples of

Jacobi form in many variables and

I would like to continue the series of examples of this type.

So examples.

Of Jacobi forms in many variables.

First of all, we consider the lattice D m.

I gave you the definition, but I repeat it once more.

This is even sublattice

The even sub lattice of the simplest Euclidean lattice of rank m.

The sum of coordinate is even modular 2.

Sub lattice of index 2 and z m.

And for those of you who worked with the algebra and

with root system, in my course I use Dm for the notation of the lattice.

And then for the root system,

I will use annotation R of Dm, all roots.

This is all vectors of square 2 in our lattice.

You can find them, for example, and you can prove without any problem,

then the cardinality of the root system of Dm is equal to 2 times m(m-1).

If you haven't any experience with root system,

please find all rows of two vectors in this lattice.

So, what kind of examples of Jacobi form for

the lattice Dm, we can construct more or less for

free using only Jacobi theta series.

First of all, I would like to construct the first

and the main generator of the graded ring

of the weak Jacobi form for Dm.

It was the following function of weight -m,

and index 1.

This is a product.

Theta (z1) over eta

cube (tau) and so on.

Theta (z m) over eta cube (tau).

This is a weak Jacobi form of weight -m for the lattice Dm.

As an exercise, this is not so simple for

the moment, you can try to analyse the space.

For example try to analyse a dimension of the space.

Using theta function we can also construct holomorphic form,

for example the first function this is so-called theta series for D8,

The simple product of eight function.

It was our first example.

We started the definition of Jacobi form in many variables from this example.

This is holomorphic Jacobi form of weight 4 for

the lattice D8.

Later we'll see that this is a minimal possible weight for

holomorphic modular form.

So this weight is called the singular weight.

4 is the singular weight,

means the minimal possible.

For this lattice.

It's really the first Jacobi form for this lattice.

If we take D7, can we construct holomorphic

Jacobi form for D7 using the same procedure?

So if z has dimension 8,

now we have 7 variables.

We can do this using the same product,

but we take only 7 theta series.

But then the character, or it's better to say the multiplier system is not trivial.

We have v added to the power 21.

To kill it, we add the cube of Dedekind eta function.

So we have Jacobi cusp form of weight 5 for this D7.

And this is cusp form, cusp.

You can continue without any problem.

Please do this mathematics.

D6 We

take only 6 vectors,

And you'll have Jacobi cusp form of weight 6 for D6 and so on.

Please continue these series, moreover, now you see how you can construct

Jacobi form or Jacobi cusp form for any lattice Dm.

It means the product of Jacobi theta and

the Dedekind eta always gives you a good example.

Now, what is about another root system?

Can we use Jacobi theta series to construct

some rather fundamental functions for them?

So for example, the first function.

This is very important, because this is really the main

generator of the graded ring of the weak Jacobi form.

And this graded ring is very important in the theory of singularity and

in the theory of forbidden structure.

Now, let's analyze the lattice A n.

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